Exponential forgetting and geometric ergodicity in state-space models

In this paper, the problem of exponential forgetting and geometric ergodicity for optimal filtering in general state space models is considered. We consider here state-space models where the latent process is modeled by a Markov chain taking its values in a continuous space and the observation at each point admits a distribution dependent on both the current state of the Markov chain and the past observation. Under given regularity assumptions, we establish that: (1) the filter, and its derivatives with respect to some parameters in the model, have exponential forgetting properties; and (2) the extended Markov chain, whose components are the latent process, observation sequence, filter and its derivatives is geometrically ergodic.